Mathematics: Calculus, Algebra and Coordinate Geometry
Mathematics in JEE Advanced tests algebra, calculus, coordinate geometry, vectors and 3D geometry, probability, and trigonometry. The paper favours multi-step problems, rigorous domain checks, and questions where a clean substitution or symmetry argument saves time.
Core rules and concepts
- Limits: use standard expansions - sin x = x − x³/6 + ⋯, tan x = x + x³/3 + ⋯, eˣ = 1 + x + x²/2 + ⋯ - before applying L’Hôpital repeatedly.
- Differentiation: a function is differentiable at a only if both one-sided derivatives exist and agree.
- Definite integrals: exploit symmetry, ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx, and substitution to reduce the integrand.
- Coordinate geometry: for a circle, use the perpendicular-distance condition for tangency; for conics, use the discriminant condition carefully.
- Probability: P(A | B) = P(A ∩ B)/P(B); distinguish independent events from mutually exclusive events.
Worked example
Evaluate L = lim (x→0) (tan x − sin x)/x³.
Using expansions, tan x − sin x = (x + x³/3 + ⋯) − (x − x³/6 + ⋯) = x³/2 + O(x⁵). Therefore L = lim (x→0) [(x³/2 + O(x⁵))/x³] = 1/2. The cancellation is cubic, so differentiating once or twice would still leave indeterminate forms; the expansion exposes the leading term.
Traps that cost marks
- Applying L’Hôpital when the limit is not 0/0 or ∞/∞.
- Squaring an equation and forgetting to reject extraneous roots.
- Treating P(A ∩ B) = P(A)P(B) as universal rather than only for independence.
- Ignoring domain restrictions in inverse trigonometric functions.
- Using the wrong order in vector cross products, a × b = −b × a.